Dual-Mode Soliton Solutions for Simplified Modified Camassa Holm and Gardner Equations

Abstract

The concept of dual-mode equations focuses on nonlinear models that describe the movement of waves in two directions at the same time, influenced by how their phase changes. Initially, Korsunsky introduced a dual-mode model that improved the Ko rteweg De Vries equation (KDVe) by turning it into a second-order formulation. In our research, our aim is to develop a new and simpler dual-mode model, which is based on the Simplified Modified Camassa-Holm (SMCH) model and the Gardner equation derived from an ideal fluid model. We will then systematically analyze its solutions and explore its geometric characteristics. To find analytical solutions for these bidirectional waves, we’ll use various methods like the extended exponential function expansion, ra tional sine/cosine method, tanh/coth method, direct approach of tan/cot method and Kudryashov method. Furthermore, we will conduct a thorough examination of how the speed at which the phase changes affects the behavior of these dual waves, using both 2D and 3D graphs. The solutions we obtain in this study have significant implications for our understanding of how solitons propagate in the field of nonlinear optics. These models can be used in many different fields and the solutions we obtain will help make it clearer how various non-linear phenomena work. This includes areas like plasma physics, Bose-Einstein condensates, shallow water waves and more.